Algebraic Curves: An Introduction to Algebraic GeometryBenjamin, 1969 - 226 pages |
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Page 53
... comaximal if I + J = R , i.e. if 1 = a + b , a ε I , bε J. For example , two distinct maximal ideals are comaximal . LEMMA . ( 1 ) IJ C IN J for any ideals I , J . ( 2 ) If I and J are comaximal , IJ = IN J. Proof : ( 1 ) is trivial ...
... comaximal if I + J = R , i.e. if 1 = a + b , a ε I , bε J. For example , two distinct maximal ideals are comaximal . LEMMA . ( 1 ) IJ C IN J for any ideals I , J . ( 2 ) If I and J are comaximal , IJ = IN J. Proof : ( 1 ) is trivial ...
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... comaximal ideals in R. 2 m Show that I + J = R. Show that I ** and comaximal for all and J i m , n . are ( b ) Suppose I , ... , IN are ideals in R , and I ; ΠΙ = j are comaximal for all i . Show that ji In ... IN = ( 11 ' • ... .. IN ) ...
... comaximal ideals in R. 2 m Show that I + J = R. Show that I ** and comaximal for all and J i m , n . are ( b ) Suppose I , ... , IN are ideals in R , and I ; ΠΙ = j are comaximal for all i . Show that ji In ... IN = ( 11 ' • ... .. IN ) ...
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... comaximal ( Problem 2-45 ) , it follows ( Problem 2-40 ) that ( 1 ) F. ( P. ) j E = 1 i = ( I1 ) I. ( Î ̧ ́ ...... . • ï ̧ ) a = ( ~ 1 ̧ ) a c I. I. ... Now choose Fε k [ X1 , ... , X ] such that F. i = = 0 if i # j , F1 ( P1 ) = 1 - i ( 1 ...
... comaximal ( Problem 2-45 ) , it follows ( Problem 2-40 ) that ( 1 ) F. ( P. ) j E = 1 i = ( I1 ) I. ( Î ̧ ́ ...... . • ï ̧ ) a = ( ~ 1 ̧ ) a c I. I. ... Now choose Fε k [ X1 , ... , X ] such that F. i = = 0 if i # j , F1 ( P1 ) = 1 - i ( 1 ...
Table des matières
Chapter One Affine Algebraic Sets 1 Algebraic Preliminaries | 1 |
Affine Space and Algebraic Sets | 7 |
The Ideal of a Set of Points | 10 |
Droits d'auteur | |
26 autres sections non affichées
Autres éditions - Tout afficher
Expressions et termes fréquents
affine variety algebraic set algebraic subset algebraically closed assume Bezout's Theorem birational birationally equivalent called change of coordinates Chapter closed subvariety comaximal COROLLARY curve F curve of degree defined deg(D deg(F denote div(G div(z divisor element F and G f ɛ finite number follows form of degree function field Hint homogeneous hyperplane hypersurface induced integer intersection number isomorphism k[X₁ LEMMA Let F linear local ring maximal ideal module-finite morphism mp(F Noether's non-singular non-zero Nullstellensatz Op(C Op(V open subvariety ordinary multiple points plane curve point on F polynomial map Problem projective change projective curve projective plane curve projective variety Proof Prop PROPOSITION quadratic transformation quotient field R-module rational function residue ring homomorphism simple point subring subvariety Suppose tangent line uniformizing parameter unique V₁ vector space Xn+1 zero Ρε